This module presents a type class for numbers which have
representations for transfinite values. The idea originated from
the IEEE-754 floating-point special values, used by
Data.Number.LogFloat. However not all Fractional types
necessarily support transfinite values. In particular, Ratio
types including Rational do not have portable representations.

For the Glasgow compiler (GHC 6.8.2), GHC.Real defines 1%0
and 0%0 as representations for infinity and notANumber,
but most operations on them will raise exceptions. If toRational
is used on an infinite floating value, the result is a rational
with a numerator sufficiently large that it will overflow when
converted back to a Double. If used on NaN, the result would
buggily convert back as negativeInfinity. For more discussion
on why this approach is problematic, see:

Documentation

Many numbers are not Bounded yet, even though they can
represent arbitrarily large values, they are not necessarily
able to represent transfinite values such as infinity itself.
This class is for types which are capable of representing such
values. Notably, this class does not require the type to be
Fractional nor Floating since integral types could also have
representations for transfinite values. By popular demand the
Num restriction has been lifted as well, due to complications
of defining Show or Eq for some types.

In particular, this class extends the ordered projection to have
a maximum value infinity and a minimum value negativeInfinity,
as well as an exceptional value notANumber. All the natural
laws regarding infinity and negativeInfinity should pertain.
(Some of these are discussed below.)

A transfinite value which is greater than all finite values.
Adding or subtracting any finite value is a no-op. As is
multiplying by any non-zero positive value (including
infinity), and dividing by any positive finite value. Also
obeys the law negate infinity = negativeInfinity with all
appropriate ramifications.

Additionally, any mathematical operations on notANumber
must also return notANumber, and any equality or ordering
comparison on notANumber must return False. Since it
returns false for equality, there may be more than one machine
representation of this value.

Since the normal log throws an error on zero, we
have to redefine it in order for things to work right. Arguing
from limits we can see that log 0 == negativeInfinity. Newer
versions of GHC have this behavior already, but older versions
and Hugs do not.

This function will raise an error when taking the log of negative
numbers, rather than returning notANumber as the newer GHC
implementation does. The reason being that typically this is a
logical error, and notANumber allows the error to propegate
silently.

In order to improve portability, the Transfinite class is
required to indicate that the Floating type does in fact have
a representation for negative infinity. Both native floating
types (Double and Float) are supported. If you define your
own instance of Transfinite, verify the above equation holds
for your 0 and negativeInfinity. If it doesn't, then you
should avoid importing our log and will probably want converters
to handle the discrepancy.

The realToFrac function is defined to pivot through
a Rational according to the haskell98 spec. This is non-portable
and problematic as discussed above. Since there is some resistance
to breaking from the spec, this class defines a reasonable variant
which deals with transfinite values appropriately.

N.B. The generic instance for transfinite types uses expensive
checks to ensure correctness. On GHC there are specialized
versions which use primitive converters instead. These instances
are hidden from other compilers by the CPP. Be warned that the
instances are overlapped, so you'll need to give type signatures
if the arguments to realToFrac are polymorphic.

If any of these restrictions (CPP, GHC-only, OverlappingInstances)
are onerous to you, contact the maintainer (we like patches :)